Forward Scattering - The Weblog of Nicholas Chapman

The weblog of Nicholas Chapman

Intuitive Quantum Electrodynamics<br>Posted 4 Jul 2026

Visual summary of some of the core ideas of Quantum Electrodynamics.<br>The wavepacket of a matter particle has a rotation in the complex plane. This couples with an internal rotation of the electromagnetic field, encouraging its rotation. This rotation propagates outwards in space. This EM field rotation in turn causes phase differences across other matter wavepackets, causing them to accelerate towards or away from the original particle. This is the electric force! Read on for the details.

Matter

Quantum electrodynamics describes the behaviour of electrically charged matter (e.g. electrons) and its interaction with the electromagnetic field.

Consider electrons - they are a particle that have a property called spin. In particular they are spin-1/2 particles. Spin is a somewhat mysterious property of particles that we won't go into too much here.

Anyway, spin 1/2 particles such as the electron are called fermions, and can be described with the Dirac equation.

Understanding the Dirac equation is pretty tricky however - it's a weird equation which Dirac came up with to get an equation that is first order in time (meaning it has first derivatives with respect to time, but not second or higher derivatives).

The Dirac equation is similar to a wave equation, and as you will see, particles described by the Dirac equation (e.g. electrons) have very wave-like behaviour.

Let's see some simulations of various solutions of the Dirac equation:

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A wavepacket moving slowly to the right (positive momentum).

The Dirac equation is an equation for a function named \( \psi \). \(\psi\) is function of position and time coordinates: \(\psi(r, t)\). It's a function from space and time coordinates to a 4-vector of complex values.<br>Since there are 3 space coordinates and 1 time coordinate, therefore it has signature

$$<br>\psi : \mathbb{R}^4 \to \mathbb{C}^4<br>$$

From a programmer's point of view, if you were to simulate a single particle on a grid using the Dirac equation, this means that at each grid cell you store 4 complex numbers. And each complex number is just 2 real numbers.

Let's take a look at a stationary particle, described using a stationary gaussian wavepacket. This particle has zero momentum.

A stationary particle/wavepacket.

Some important things to note about the stationary particle simulation video: In the bottom middle section, you will see some phasor arrows rotating clockwise. Phasor arrows show the complex value at a bunch of spatial points. Only the red arrows are non-zero length here: this means that \(\psi_1\) is non-zero, the other \(\psi\) components are (nearly) zero.

The top-right section shows the momentum density G, from which you can see the electron spin direction (out of the screen in this case along the +z axis). Note however that the electron spin direction is *not* correlated with the direction of rotation of the complex phasor values.

To reinforce that point, here's a video of a spin-down particle, e.g. one whose spin vector points in the -z direction (into the screen). Note that the momentum density curls in the opposite direction, but the phasor arrows rotate in the same clockwise direction as for the spin-up particle.

A stationary particle/wavepacket with spin down (-z).

Now let's look at anti-matter! These solutions are referred to as negative energy solutions:

A stationary negative energy (antimatter) particle/wavepacket

The crucial difference visible in the antimatter video above is that compared to normal matter, the phasor arrows spin in the opposite direction: anti-clockwise . Indeed, that's basically what antimatter is: it's a solution to the Dirac equation where the rotation direction in the complex plane goes the opposite direction from normal matter.

The effect of matter on the electromagnetic field

In standard quantum electrodynamics, we have a value called \( \rho \), which is the probability density of \(\psi\) at a particular position and time. For the Dirac equation,

$$<br>\rho = \psi^\dagger \psi = |\psi_1|^2 + |\psi_2|^2 + |\psi_3|^2 + |\psi_4|^2<br>$$<br>In other words, the probability density \(\rho\) is equal to the sum of the squared lengths of the psi vectors in the complex plane.

This is what is drawn in the bottom left section of the videos above. Note that \(\rho\) is greater than or equal to zero for both the matter and antimatter particles!

The probability density \(\rho\) then affects \(\Phi\), the electric potential, in the following way (in the Lorenz gauge):

$$<br>(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2) \Phi = q \rho<br>$$<br>Where \(q\) is the electric charge of the particle, and has magnitude equal to the coupling constant between the dirac and EM field.

Rearranging:

$$<br>\begin{aligned}<br>\frac{1}{c^2} \frac{\partial^2}{\partial t^2} \Phi = q \rho + \nabla^2 \Phi \\<br>\frac{\partial^2}{\partial t^2}...